Entropic Probability Conservation and the Decomposition Po(t)+Pe(t)=1

A central structural feature of the Theory of Entropicity (ToE) is the division of physical evolution into two orthogonal sectors: the observer (coherent) sector and the entropic sector. This division is not merely conceptual; it is encoded directly in the Hilbert‑space architecture of the theory and leads to a distinct probability‑conservation law that differs from the classical Kolmogorov formulation.

To formalize this structure, the total Hilbert space is decomposed as

$${\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,$$

where ${\mathcal{H}}_o$ represents the coherent observer sector and ${\mathcal{H}}_e$ represents the entropic sector. The ToE evolution operator acts on the total state through a combined unitary–entropic flow,

$$U_{\mathrm{T}\mathrm{o}\mathrm{E}}(t)=e^{-iHt}{e}^{-Ct},$$

where $H$ generates coherent evolution and $C$ generates entropic dissipation. Under this evolution, the total state decomposes as

$$\mathrm{\Psi }(t)={\psi }_o(t)+{\psi }_e(t),$$

with the orthogonality condition

$${\psi }_o(t)\perp {\psi }_e(t)\mathrm{.}$$

Norm conservation of the total state,

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=1,$$

implies the additive relation

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

Defining the sectoral probabilities as

$$P_o(t):=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2,P_e(t):=\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2,$$

one obtains the entropic probability‑conservation law,

$$P_o(t)+P_e(t)=1.$$

This relation is not a restatement of the classical normalization axiom ${\sum }_i{P}_i=1$. Instead, it expresses a binary partition of the total quantum state into two dynamically coupled but orthogonal sectors. Classical probability theory partitions events; the Theory of Entropicity partitions sectors of physical reality. The observer sector captures coherent, information‑bearing evolution, while the entropic sector captures the irreversible flow of amplitude into the informationally inaccessible domain generated by $C$.

Thus, the equation

$$P_o(t)+P_e(t)=1$$

is a conservation law arising from the Hilbert‑space structure of ToE and the combined unitary–entropic dynamics. It encodes the fundamental principle that while amplitude may flow from the observer sector into the entropic sector, the total probability remains conserved across the full ToE evolution. This decomposition provides the mathematical foundation for entropic irreversibility, observer‑dependent coherence, and the emergence of classicality within the ToE framework.

Scholium

The ToE probability law is about sectoral decomposition, not human observation

The law:

$$P_o(t)+P_e(t)=1$$

is not about what a person sees. It is about how the universe partitions amplitude between:

• the coherent (observer) sector

• the entropic sector

This is why ToE emphasises:

“${\psi }_o(t)\perp {\psi }_e(t)$” “${\mathcal{H}}_{\text{tot}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e$” “$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2$”

This is geometry, not psychology.

Reference

1.       J. O. Obidi, "On the Discovery of New Laws of Conservation and Uncertainty, Probability and CPT-Theorem Symmetry-Breaking in the Standard Model of Particle Physics: More Revolutionary Insights from the Theory of Entropicity (ToE)," Cambridge University (CoE), 2025.  https://doi.org/10.33774/coe-2025-n4n45

2.   https://theoryofentropicity.blogspot.com/2026/04/entropic-probability-conservation-and.html

3.    https://theoryofentropicity.blogspot.com/2026/04/probability-as-conservation-law-in.html

4.    https://theoryofentropicity.blogspot.com/2026/04/scholium-sectoral-probability.html